Mathematical Methods in the Applied Sciences · 2018 · 37 citations · 20 references
Natural SciencesMathematicsPhysicsComputational Fluid DynamicsDynamical SystemsFluid DynamicsCompressible FlowDissipative SystemsFlow StabilityNavier-stokes EquationsAllen-cahn EquationCauchy ProblemCompressible Navier-stokes EquationsRarefaction WavesStability Criteria In Waves
In this paper, we are concerned with the large time behavior of solutions to the Cauchy problem for the one dimensional Navier‐Stokes/Allen‐Cahn system. Motivated by the relationship between the Navier‐Stokes/Allen‐Cahn system and the Navier‐Stokes system, we can prove that the solutions to the one‐dimensional compressible Navier‐Stokes/Allen‐Cahn system tend time‐asymptotically to the rarefaction wave, where the strength of the rarefaction wave is not required to be small. The proof is mainly based on a basic energy method.
20